carga linear e não linear em sistemas monofásicos
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CURRENT HARM ONICS ANALYSIS OF NON-LINEAR SINGLE-PHASE LO ADS
IN A THREE-PHASE NETWORK
V . Spitsa and A . Alexandrovitz
Department of Electrical Engineering,
Technion-Israel Institute of Technology,
Haifa 32000, Israel
A B S T R A C T
This paper presents results of the current harmonics
analysis in a three-phase network with nonlinear single-
phase loads. In this analys is, typical single-phase nonlinear
loads such as computers, m onitors and printers, were
considered. Simulation results are accompanied with field
measurements. A mutual influence of single-phase loads
connected in different phases .o f the network is studied.
Factors, having an effect on harmonic content of a neutralconductor current, are investigated in the present work.
The c auses for the excessive neutral conductor current are
explored and the dominant role of the triplen harmonics is
highlighted.
1 . I N T R O D U C T I O N
Single-phase nonlinear loads, such as computers, TV sets.
fluorescent lightning and different electronic devices are
the main source of harmonics in commercial and
residential building networks. I n recent years their amount
grows rapidly leading to excessive neutral conductorcurrents [2],[3],[7]. As a result, harmonic analysis of the
single-phase nonlinear load s has an increasing importance.
Pom,er supplies of the single-phase nonlinear loads are
usually implemented according to a switch-mode scheme
with dio de-b ridg e rectifier presented in Fig. I . Its typical
harmonic spectrum consists of odd harmonics only. The
low-order harmonics in this spectrum have especially high
magnitudes which causes a highly non-sinusoidal current
waveform. Influence of the single-phase diode-bridge
rectifier harmonics on electrical network performance and
factors affecting magnitude levels of these harmonlcs are
studied in the present paper. A special attention is devoted
to the problem of the excessive neutral conductor current
resulted from triple harmon ics coupling.
078034427-wO~20.0~2004EEE
Fig. 1. Single-phase switch-mode pow er supp ly.
2. A N A L Y S IS M E T H O D
Single-phase diode bridge rectifier operation can be
described by a se t of differential equa tions with particular
continuity conditions [4]-[6]. As a result, current and
voltage waveforms of the rectifier can be obtained using
the strategy proposed in [ I ] , which exploits a classical
solution method of differential equations. A novel
development presented in this paper is based on
incorporation o f the above mentioned dio de-bridge
rectifier model into a composite load of three-phase
electrical network with neutral conductor. For this
purpose, time-domain simulations were performed using
MatlabiSimulink package. A symmetric sinusoidal three-
phase voltage source was assumed. The obtained phase
currents can be written in the following manner:
iH =ti;z&Ihcos[h . (w ' t 1 20")+9.1 (2 )h= l h=l
m
i1. =ti: c&Ihcos[ h . ,It +120")+p h] (3)h=I h=
where
i A , i H , i c are phase currents in phases A , B and C
respectively.
r A , r B , r l . are currents of the particular harmonic o f order
h in phases A, B an d C respectively,
.I! .I, .I?
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I h is a currentR M S value of particular harmonic oforder
p h i s a current phase of particular harmonic oforder h ,
m' is a fundamental angular frequency,
1 is a time instant.
These current waveforms and their harmonic spectrumare analyzed using two parameters. The first one is an
R M S value of he harmonic current:
h ,
*"ib, h . l m
R ~ r e Xwim icI$hh nm
,I A A A 4 m -iN-
(4)v h= l
The second parameter is a total harmonic distortion (THD)
definedas
The obtained results are described in the next sections.
3. S I N G L E - P H A S E L O A D S I M U L A T I O N R E S U L T S
A simulation of the single-phase diode-bridge rectifier
operation was performed according to the electrical
scheme shown in Fig. I , where the rectifier load
represented by a constant resistance Rdc. Parameters o f
the scheme elements are given in Appendix. The obtained
input current waveform and its harmonic spectrum are
presented in Fig. 2. An RMS value of this current is
0.929[A] while a magnitude of its fundamental component
is 0.431[A]. I t is obvious that current THD value of
190.82% is extremely high in the present case. This factresults from an existence of low order harmonics with
significant magnitudes. Itshould be noted thata magnitude
of the third harmonic is almost equal to the fundamental
current component. I t will be shown later that this
phenomenon is of special importance in three-phase
electrical networks with neutral conductor.
4. S I M U L A T I O N R E S U L T S OF T H R E E - P H A S E
E L E C T R I C A L N E T W O R K
A three-phase electrical network with neutral conductor
and single-phase non-linear loads is investigated in the
present section. The equivalent scheme of this network is
shown in Fig. 3.
Inspecting an order of the current phase interchangeaccording to ( I ) , one can conclude that harmonics of order
3k+, where k=0,1,2, ..., are composing a positive
sequence system, harmonics o f order 3k+2 are forming a
negative sequence system and harmonics of order 3k are
Fig. 2. Input current ofthe single-phase diodebridge rectifier.
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phase currents. As a result, the neutral conductor is in
more severe conditions than the phase conductors and a
potential danger of its overheat and damage is present
even in the totally symmetric balanced network. Therefore,
a special care should be taken whena cross section area of
the neutral conductor is determined.
5. E X P E R I M E N TA L R E S U L T S
Computer simulations and harmonic analysis using
analytical models are based on assumptions and
simplifications, which are made during solution stages.
Moreover, exhaustive data about an electrical network
contiguration are not always available. Therefore
experimental measurements are very useful for model
validation and determination of harmonics levels in
electrical network elements. In the present section.
experimental results of the harmonic measurements are
given. These results allow one to verify a qualitative
content of harmonic spectrum of the phase and neutral
conductor currents in a three-phase electrical network,which was obtained analytically. A l l measurements were
carried out using power analyzers "Voltech PM100 and
"Fluke 418" in the Energy Conversion Laboratory ofthe
Electrical Engineering Department in Technion.
5.1. C o m p u t e r l o a d h a rm o n i c s .
Computers are a major part of the single-phase
nonlinear loads in the commercial and residential electrical
networks. Therefore, reliable infohation on computer
load harmonics has a primary importance for harmonic
analysis o fthe above mentioned electrical networks. In the
present subsection, measurement results of the computer
load harmonics are given.The computer load is typically a unit consisting of the
following elements: tower, monitor and printer. The
measured waveform and harmonic spectrum of these loads
are shown in Fig. 6-8 respectively. A total current of the
computer unit is treated inFig. 9.
5.2 C u r r e n t h a rm o n i c s in a t h ree-phase e lec t r i ca l
n e t w o r k w i t h n e u t r al c o n d u c to r
The results of the current harmonics measurements at
the input of the distributing board in the PC-farm of the
Electrical Engineering Department are presented in the
present subsection. A three-phase line and neutral
conductor currents are taken into consideration. lheirharmonic spectrums and waveforms are shown in Fig. 10-
13. Comparing Fig. 10-12, it can be concluded that a load
distribution among the phases is unbalanced. Indeed.
Phase Conductor Current Waveform
0.02 Time, [sec] 0.04 0.W
Harmonic Spectrum of Phase Conductor Cur rent
z 0.43- I ,
I .3 -
i! 0.14-'i .M- 1L5 7 9 11 13 15 17 19 21 23 25 27 29 31 33
HamO" i c order
Fig. 4. Harmo nic spectrum of the phase conductor current s.
Neutral Conductor Current Waveform
0.02 Time, [sec] 0.04 0.W
Harmonic Spectrum of Neutral Conduc tor Current
z 1.26- , 10.43-
0 3 15 21 27 33
Fig. 5. Harmonic spectrum of the n eutral conductor cur rent .
Fig. 6. Input current harmonics and waveform
of "Mediatek" computer tower.
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waveforms o f the currents in phases A and B have a bell-
shape form, which is typical for the computer loads, while
a waveform of the current in phase C is almost sinusoidal.
Hence, a majority o f electrical loads in phase C is linear
(air conditioners). Neutral conductor current is a
geometric sum of the phase currents. Consequently, it
represents an unbalanced part of phase currents togetherwith their triplen harmonic content as it is shown in Fig.
13. In this figure a fundamental component of the
harmonic spectrum has a value, which is almost equal to
the fundamental component of the line current in phase C
which is depicted in Fig. 12. This fact indicates that there
exists a very large unbalance of load powers among the
phases of the electrical network. Moreover, uneven
distribution of the single-phase nonlinear loads between
the phases of the electrical network causes an additional
enhancement of the neutral conductor current due to the
fact that first and fifih-order phase current harmonics are
not compensated in this case.
The third-order current harmonics forma zero-sequence
system, they do not cancel in the neutral wire. This is a
primary reason for a high neutral current in the three-phase
systems with the single-phase rectifier loads. In the present
subsection, a third harmonic o f the neutral conductor
current shown in Fig. 13, has a value. which is 78% of the
current fundamental component in phase B,given in Fig.
11. As a result, the neutral current has the RM S value,
which is greater than the R M S value o f the current in
phaseB.
6. CONCLUSION
In the present paper current harmonics produced by the
single-phase rectifier loads in a three-phase electrical
network were studied. The effects resulted from the
diversity o f the load parameters of these loads were
investigated. The simulation results were verified by
measurements. The main factors. which lead to an
excessive neutral conductor current, were determined. .They are an unbalance of oad powers installed in different
phases, a diversity of the current harmonic spectrums of
the phase loads and a presence of the significant triplen
harmonics in the phase currents produced by single-phase
nonlinear loads. Itwas obtained that RMS valueof neutral
conductor current may be higher than that of phase
currents. Therefore, it is recommended to choose a cross
section area of neutral conductor to be not less thana cross
section area of phase conductors in the three-phaseelectrical networks with a large number of the single-phase
rectifier loads.
Fig. 7. Input current harmonics and waveform
of "Packard Bell" monitor.
m-r c u m W*"Cform
I .U- I _ .I< 1117 1111 .* ID. .P _o I Y I .0 0 0 ,m ,m m -,U ,U
n--1.
Fig. 8. Input current harmonics and waveform
of "HP Laser Jet Illp " printer.
Fig. 9. Input current harmonics and waveform
of a typical computer unit.
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HarmonicSpeCtNmofPhaseA Current HarmonicSpectrum ofPhaseB Current
.~: ,:.
C U m o . l * l .,
. .
7 , ,,,, ..- ,:., ,wll , ~ .,. .
currm.lAl , --.~,
'. ... i
. ~,.
I ..I \ ~ . ~
. .
-:,, ..,: 1,1 (,," -El. ,~.._.L-.~-
'\ iY
Phase C Current Waveform
C Y r n . l * l .,
T,"E. I-,
Fig. 12. Current in phase C o f the input three-phase line.
c"mm.lA, ,,
:,
10. APPENDIX
Supply voltage: V ' = 2 2 O [ V ] , f' 50 [H;]
Parametersof
electrical wire:c, 0 . 2 [ ~ 1 , L, ,~0 . 4 [ 4
R, , =o.ols[n], L,, = l [ p H ]
Parameters of diodes:
Parameters of the diode-bridge rectitiers
Rdc =lOOO[n], C,,' = 3 7 0 [ p F ]
11. REFERENCES
,'.. , -7 ,\
1 , ",,I : ~ , pI,k~r
.-/ I '
L.'. . ~ . ... . .~.. ~ ~ ~ ~ ~
[ I ] 0. Boix, L. Sainz and J. Pedra, "Harmonic interaction in
capacitor rectifier loads", ETEP, vol. 10. no. 2. MarcW April.2000.
[2] J. 1. M. Desmet. 1. Sweettvaegher. G. Vanalme. K. Stocman
and R. J. M. Belmans, "Analysis of the neutral conductor cumentin a three-phase supplied network with nonlinear single-phaseloads", IEEE Transactions on lndustw Applications.vol. 39, no.3, pp. 587-593, May/June 2003.
333